3 Persons are tagged with #maths

Refurio Anachro
Refurio Anachro
refurioanachro@diaspora.glasswings.com
#maths #fractalart #unix #philosophy

#maths

clarice@diaspora.glasswings.com
clarice oncemore

"Neil DeGrasse Tyson Talks about how the United States is being left behind in areas such as physics, math, and engineering comparing it to Europe and Japan which are generating more scientific papers and have better infrastructure.
Drawing upon his deep understanding of scientific trends and global dynamics, Tyson highlights the contrasting landscapes of scientific progress between the United States, Europe, and Japan. He articulates how these regions have surged ahead, not only by generating more scientific papers but also by investing in cutting-edge infrastructure and fostering collaborative research environments.

"..Throughout the video, Tyson engages in a captivating analysis of the key elements that have propelled Europe and Japan to the forefront of scientific innovation. He discusses the significance of robust funding mechanisms, visionary leadership, and educational strategies that cultivate a new generation of researchers and innovators. With an analytical yet accessible approach, Tyson addresses the underlying challenges that the United States must confront to regain its competitive edge in the global scientific arena."
https://www.youtube.com/watch?v=HgmeoDQWGLQ

#maths #science #entertaining

2 Likes
1 Comments
tord_dellsen@diasp.eu
Sunyata ☸ 👨🏻‍💻

#JustForFun i created these Sierpinski triangles using #Python's #turtle module | #maths #math #programming #fractals #recursion

import logging
import turtle
import time

SPEED_INT = 5  # 1-10 where 0 is as fast as possible
logging.basicConfig(level=logging.DEBUG)

DISPLAY_WIDTH = 1620
# Turtle setup
screen = turtle.getscreen()
t = turtle.getturtle()
turtle.title("Sierpinski Triangle (using turtle)")
t.shape("turtle")
t.speed(SPEED_INT)
turtle.setheading(0)
# ..set start position
t.penup()
turtle.setx(-DISPLAY_WIDTH // 2)
turtle.sety(-450)
t.pendown()
start_pos = t.pos()

time.sleep(1)


def draw_recursive(i_level: int, i_direction: int):
    """Level 1 is the starting i_starting_level (0 doesn't exist)"""
    logging.debug(f"{i_level * '-'} draw_recursive called with level {i_level}")
    if i_level == 1:
        t.forward(distance)
        return
    draw_recursive(i_level - 1, -i_direction)
    t.left(i_direction * 60)
    draw_recursive(i_level - 1, i_direction)
    t.left(i_direction * 60)
    draw_recursive(i_level - 1, -i_direction)


distance = 0


def draw_(i_starting_level: int, i_pen_color_channel: float = 0.0):
    logging.debug(f"==== draw_ called with starting level {i_starting_level} ====")
    t.penup()
    t.setpos(start_pos)
    t.setheading(0)
    t.pendown()

    global distance
    distance = 1024 / (2 ** (i_starting_level - 1))
    distance = min(100, distance)

    pen_size = distance // 2
    if pen_size < 1:
        raise Exception("Pen size is smaller than one")
    t.pensize(distance // 2)

    t.pencolor((i_pen_color_channel,) * 3)

    if i_starting_level % 2 == 0:
        direction = 1
    else:
        direction = -1
    draw_recursive(i_starting_level, direction)


# draw_(9)

pen_color_channel = 0.8
for i_starting_level in range(5, 8):
    if pen_color_channel < 0:
        raise Exception("Pen color channel smaller than zero")
    draw_(i_starting_level, pen_color_channel)
    pen_color_channel -= 0.2

t.hideturtle()
turtle.done()
5 Likes
2 Comments
1 Shares
tord_dellsen@diasp.eu
Sunyata ☸ 👨🏻‍💻

#JustForFun i created these #KochSnowflake/s using #Python's #turtle module | #maths #math #programming #fractals #recursion

import logging
import turtle
import time

SPEED_INT = 4  # 1-10 where 0 is as fast as possible
logging.basicConfig(level=logging.DEBUG)

DISPLAY_WIDTH = 1620
# Turtle setup
screen = turtle.getscreen()
t = turtle.getturtle()
turtle.title("Koch Snowflake")
t.shape("turtle")
t.speed(SPEED_INT)
turtle.setheading(0)
# ..set start position
t.penup()
turtle.setx(-DISPLAY_WIDTH // 2)
turtle.sety(-400)
t.pendown()
start_pos = t.pos()

time.sleep(1)


def draw_recursive(i_level: int):
    """Level 1 is the starting level (0 doesn't exist)
    , i_unit_distance: int
    """
    if i_level == 1:
        t.forward(distance)
        return
    draw_recursive(i_level - 1)
    t.left(60)
    draw_recursive(i_level - 1)
    t.right(120)
    draw_recursive(i_level - 1)
    t.left(60)
    draw_recursive(i_level - 1)


def get_distance(i_levels_to_draw):
    distance_ = DISPLAY_WIDTH / (3 ** (i_levels_to_draw - 1))
    return distance_


pen_color_channel = 0.9
pen_size = 16

for i in range(2, 7):
    t.penup()
    t.setpos(start_pos)
    t.pendown()

    levels_to_draw = i
    distance = get_distance(levels_to_draw)
    if pen_size < 1:
        raise Exception("Pen size smaller than one")
    t.pensize(pen_size)
    pen_size = pen_size // 2
    if pen_color_channel < 0:
        raise Exception("Pen color channel smaller than zero")
    t.pencolor((pen_color_channel,) * 3)
    pen_color_channel -= 0.2

    draw_recursive(levels_to_draw)

t.hideturtle()
turtle.done()
5 Likes
1 Comments
2 Shares
tord_dellsen@diasp.eu
Sunyata ☸ 👨🏻‍💻

#JustForFun i created this Hilbert space-filling curve using #Python's #turtle module | #maths #math #programming

import logging
import random
import turtle
import time

SPEED_INT = 5  # 1-10 where 0 is as fast as possible
UNIT_DISTANCE = 30
logging.basicConfig(level=logging.DEBUG)

# Turtle setup
screen = turtle.getscreen()
t = turtle.getturtle()
turtle.title("Hilbert Space-filling Curve")
t.shape("turtle")
t.speed(SPEED_INT)
t.pensize(5)
t.pencolor((0, 0, 0))
t.penup()
turtle.setx(-500)
turtle.sety(-450)
t.pendown()
start_pos = t.pos()

time.sleep(1)

CW = 1
CCW = -1


def get_angle():
    fuzzyness = random.choice(range(0, 2))
    # degrees = 90 + fuzzyness
    degrees = 90
    return degrees


color_ = (0, 0, 0)

DIRECTION_UP = True
DIRECTION_DOWN = False

color_change_direction = (DIRECTION_UP, DIRECTION_UP, DIRECTION_UP)


def change_color():
    global color_
    global color_change_direction
    index_to_change = random.choice(range(0, 3))
    channel = color_[index_to_change]
    if color_change_direction:
        channel += 0.05
    else:
        channel -= 0.05
    if channel > 1.0:
        color_change_direction = not color_change_direction
        channel = 1.0
    elif channel < 0.0:
        color_change_direction = not color_change_direction
        channel = 0.0
    new_color = turtle.pencolor()
    new_color_list = list(new_color)
    new_color_list[index_to_change] = channel
    color_ = tuple(new_color_list)
    turtle.pencolor(color_)


def draw_recursive(i_direction: int, i_level: int):
    """Level 1 is the starting level (0 doesn't exist)"""
    if i_level == 1:
        change_color()

    def right_forward():
        t.right(i_direction * get_angle())
        t.forward(UNIT_DISTANCE)

    def forward_right():
        t.forward(UNIT_DISTANCE)
        t.right(i_direction * get_angle())

    if i_level > 1: draw_recursive(-i_direction, i_level - 1)
    if i_level % 2 == 0:
        right_forward()
    else:
        forward_right()
    if i_level > 1: draw_recursive(i_direction, i_level - 1)
    if i_level % 2 == 0:
        t.right(-i_direction * get_angle())
        t.forward(UNIT_DISTANCE)
        t.right(-i_direction * get_angle())
    else:
        t.forward(UNIT_DISTANCE)
    if i_level > 1: draw_recursive(i_direction, i_level - 1)
    if i_level % 2 == 0:
        forward_right()
    else:
        right_forward()
    if i_level > 1: draw_recursive(-i_direction, i_level - 1)


turtle.setheading(90)
draw_recursive(CW, 5)

t.hideturtle()
turtle.done()
5 Likes
1 Comments
mkwadee@diasp.eu
Khurram Wadee

A few weeks ago, I posted an example of a #projectile #trajectory in a uniform #GravitationalField without #AirResistance. It's a school-level exercise but it was nice to produce an #animation for it. To model air resistance is a bit more tricky as any #fluid, including #air, is complex and does not necessarily behave as you might expect. A simple model for it might be to assume it is a #linear #viscous fluid offering resistance proportional to the velocity of the projectile, i.e. R = -k.v, where v is the velocity vector and k is a constant which dictates how viscous the fluid is.

Happily, this model has exact solutions so I didn't need to do any numerical integration to find the form of the displacement. Here are four different cases, including the "ideal" case k = 0. The others are k = 0.1, 1 and 10, with the middle one highlighted. The essential difference between the non-resistance case and the others is that horizontal velocity steadily decreases with time and decays towards zero exponentially. The vertical velocity is also reduced but still grows. The most viscous case (k = 10) shows terminal velocity behaviour very quickly and the projectile appears to be moving through treacle.

#MyWork #CCBYSA #AppliedMathematics #Maths #WxMaxima

7 Likes
7 Comments
mkwadee@diasp.eu
Khurram Wadee

I was explaining to my wife how the #velocity #components of an idealized #projectile differ. She was having some problems understanding that only the #vertical component changes and that the #horizontal component remains constant. In the end, I knocked up this little animation to tr and explain pictorially (the program in #WxMaxima is actually interactive. so you can stop it, step through it or run it backwards if that helps).

#Maths #Mathematics #Mechanics #ConstantAcceleration #Vectors #Dynamics #MyWork #CCBYSA #FreeSoftware

18 Likes
11 Comments
mkwadee@diasp.eu
Khurram Wadee

An #Exact #Formula for the #Primes: #Willans' Formula - YouTube

I didn't know this formula existed. It was published in 1964 but if you look at it carefully, it's not a formula as such, it's actually an #algorithm coded into some nifty #maths. It gets excruciatingly slow for even modest values on n, when n represents the nth prime. I've written it as a #maxima code below.

https://www.youtube.com/watch?v=j5s0h42GfvM

#WillansFormula #Mathematics #PrimeNumbers

prime(n):=1+sum(floor((n/sum(floor(cos(%pi*(factorial(j-1)+1)/j)^2),j,1,i))^(1/n)),i,1,2^n);
6 Likes
8 Comments